482 On the Integration of Linear Differential Equations. 

 it becomes 



which may be treated in Mr. Boole's way. 



These examples may serve to indicate the mode of procedure 

 in other cases ; they belong each to a distinct class, but the 

 mode of reduction is the same. We have employed only one of 

 the forms assumed for u ; in other cases it may be necessary to 

 employ the other, and cases may perhaps arise in which it will 

 be necessary to use a combination of both forma. 

 Now let 



X = «r('sr + a)w-f A(CT—ir)('ar-ti)D*'w. , . (5) 

 Assume 



w=;('cr— r)('cr— 2r) .... (w— zV)wi. 

 Then 



D'M=D''(txr-r)...(w-zV)Mi = txr(«r-r)...(w-(i-l)r)D^Wiby(C), 



Substituting these values in the given equation, and operating 

 with the inverse of the factors common to the terms of the second 

 member, we have 



Xi = {'aT-\-a)uj-\-k{'ST + b)'D\. . 



The solution of the given equation is therefore made to depend 

 upon that of one an order lower. 

 In the last place, let 



X=^(«r-f r)w + X;(^ +a)('cr + a + (2i + l)r)D2'-M, . (6) 

 Make 



w = (tsr + fl)(«r+a-l-2r) (w + «+ (2i—2)r)t«i. 



Then by (C), 



D^M=(OT + a + 2r) .... {m + a-{-2ir)D^\, 

 And proceeding as before, we find 

 X, = 'ar(txr + r)Mi-f ^(txr + fl + 2fr)(CT + a+(2? + l)r)D^\. 

 Xi=('«r + a)-i .... (w + a + (2z-2)r)-'X. 



This by (C), in the same manner as in the second example, may 

 be put under the form 



Mate 



then 



Xj=«i+^^'m,. 



